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Main Author:	Lang, Serge 1927-2005 (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Springer 1985
Edition:	2. ed.
Series:	Graduate texts in mathematics 103
Subjects:	Analyse (wiskunde) Analyse mathématique Complexe variabelen Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis Funktionentheorie Einführung
Online Access:	Inhaltsverzeichnis
Physical Description:	XIV, 367 S. graph. Darst.
ISBN:	0387960856 3540960856

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MARC


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adam_text	Contents Prerequisites ix PART ONE Basic Theory 1 CHAPTER I Complex Numbers and Functions 3 §1. Definition 3 §2. Polar Form 8 §3. Complex Valued Functions 12 §4. Limits and Compact Sets 17 Compact Sets 21 §5. Complex Differentiability 28 §6. The Cauchy Riemann Equations 32 §7. Angles Under Holomorphic Maps 34 CHAPTER II Power Series 38 §1. Formal Power Series 38 §2. Convergent Power Series 49 §3. Relations Between Formal and Convergent Series 62 Sums and Products 62 Quotients 66 Composition of Series 67 §4. Analytic Functions 69 §5. The Inverse and Open Mapping Theorems 72 §6. The Local Maximum Modulus Principle 79 §7. Differentiation of Power Series 82 Xii CONTENTS CHAPTER III Cauchy s Theorem, First Part 87 §1. Holomorphic Functions on Connected Sets 87 Appendix: Connectedness 93 §2. Integrals Over Paths 94 §3. Local Primitive for a Holomorphic Function 103 §4. Another Description of the Integral Along a Path 109 §5. The Homotopy Form of Cauchy s Theorem 115 §6. Existence of Global Primitives. Definition of the Logarithm 118 CHAPTER IV Cauchy s Theorem, Second Part 123 §1. The Winding Number 124 §2. Statement of Cauchy s Theorem 128 §3. Artin s Proof 137 CHAPTER V Applications of Cauchy s Integral Formula 144 §1. Cauchy s Integral Formula on a Disc 144 §2. Laurent Series 151 §3. Isolated Singularities 155 Removable Singularities 155 Poles 156 Essential Singularities 158 §4. Dixon s Proof of Cauchy s Theorem 162 CHAPTER VI Calculus of Residues 165 §1. The Residue Formula 165 §2. Evaluation of Definite Integrals 180 Fourier Transforms 182 Trigonometric Integrals 185 Mellin Transforms 187 CHAPTER VII Conformal Mappings 196 §1. Schwarz Lemma 198 §2. Analytic Automorphisms of the Disc 200 §3. The Upper Half Plane 203 §4. Other Examples 206 §5. Fractional Linear Transformations 215 CONTENTS xiii CHAPTER VIII Harmonic Functions 224 §1. Definition 224 Application: Perpendicularity 228 Application: Flow Lines 230 §2. Examples 234 §3. Basic Properties of Harmonic Functions 241 §4. Construction of Harmonic Functions 244 §5. The Poisson Representation 249 PART TWO Various Analytic Topics 253 CHAPTER IX Applications of the Maximum Modulus Principle 255 §1. The Effect of Zeros, Jensen Schwarz Lemma 255 §2. The Effect of Small Derivatives 260 Hermite Interpolation Formula 261 §3. Entire Functions with Rational Values 262 §4. The Phragmen Lindelof and Hadamard Theorems 268 §5. Bounds by the Real Part, Borel Carath odory Theorem 273 CHAPTER X Entire and Meromorphic Functions 276 §1. Infinite Products 276 §2. Weierstrass Products 280 §3. Functions of Finite Order 286 §4. Meromorphic Functions, Mittag Leffler Theorem 290 CHAPTER XI Elliptic Functions 292 §1. The Liouville Theorems 292 §2. The Weierstrass Function 295 §3. The Addition Theorem 299 §4. The Sigma and Zeta Functions 302 CHAPTER XII Differentiating Under an Integral 307 §1. The Differentiation Lemma 308 §2. The Gamma Function 311 Proof of Stirling s Formula 316 Xiv CONTENTS CHAPTER XIII Analytic Continuation 324 §1. Schwarz Reflection 324 §2. Continuation Along a Path 330 CHAPTER XIV The Riemann Mapping Theorem 340 §1. Statement and Application to Picard s Theorem 340 §2. Compact Sets in Function Spaces 344 §3. Proof of the Riemann Mapping Theorem 347 §4. Behavior at the Boundary 351 Appendix Cauchy s Formula for C00 Functions 359 Index 365
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indexdate	2024-07-09T15:41:46Z
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isbn	0387960856 3540960856
language	English
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spelling	Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Complex analysis Serge Lang 2. ed. New York [u.a.] Springer 1985 XIV, 367 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 103 Analyse (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Funktionentheorie (DE-588)4018935-1 s DE-604 Funktionalanalysis (DE-588)4018916-8 s Graduate texts in mathematics 103 (DE-604)BV000000067 103 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001437325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Lang, Serge 1927-2005 Complex analysis Graduate texts in mathematics Analyse (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4018916-8 (DE-588)4018935-1 (DE-588)4151278-9
title	Complex analysis
title_auth	Complex analysis
title_exact_search	Complex analysis
title_full	Complex analysis Serge Lang
title_fullStr	Complex analysis Serge Lang
title_full_unstemmed	Complex analysis Serge Lang
title_short	Complex analysis
title_sort	complex analysis
topic	Analyse (wiskunde) gtt Analyse mathématique Complexe variabelen gtt Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis (DE-588)4018916-8 gnd Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	Analyse (wiskunde) Analyse mathématique Complexe variabelen Fonctions d'une variable complexe Functions of complex variables Mathematical analysis Funktionalanalysis Funktionentheorie Einführung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001437325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000000067
work_keys_str_mv	AT langserge complexanalysis

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